Questions: The graph of f'', the second derivative of the continuous function f, is shown above on the interval [0,9]. On this interval f has only one critical point, which occurs at x=6. Which of the following statements is true about the function f on the interval [0,9]? A f has a relative minimum at x=6 but not an absolute minimum. (B) The absolute minimum of f is at x=6. C. f has a relative maximum at x=6 but not an absolute maximum. (D) The absolute maximum of f is at x=6.

The graph of f'', the second derivative of the continuous function f, is shown above on the interval [0,9]. On this interval f has only one critical point, which occurs at x=6. Which of the following statements is true about the function f on the interval [0,9]? A f has a relative minimum at x=6 but not an absolute minimum. (B) The absolute minimum of f is at x=6. C. f has a relative maximum at x=6 but not an absolute maximum. (D) The absolute maximum of f is at x=6.
Transcript text: 9 Mark for Review The graph of $f^{\prime \prime}$, the second derivative of the continuous function $f$, is shown above on the interval $[0,9]$. On this interval $f$ has only one critical point, which occurs at $x=6$. Which of the following statements is true about the function $f$ on the interval $[0,9]$ ? A $f$ has a relative minimum at $x=6$ but not an absolute minimum. (B) The absolute minimum of $f$ is at $x=6$. C. $f$ has a relative maximum at $x=6$ but not an absolute maximum. (D) The absolute maximum of $f$ is at $x=6$.
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Solution

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Solution Steps

Step 1: Identify the critical point

The problem states that the function \( f \) has only one critical point, which occurs at \( x = 6 \).

Step 2: Analyze the second derivative at the critical point

The graph provided is of \( f'' \), the second derivative of \( f \). To determine the nature of the critical point at \( x = 6 \), we need to look at the value of \( f''(6) \).

Step 3: Determine the concavity at the critical point

From the graph, \( f''(6) \) is negative, indicating that \( f \) is concave down at \( x = 6 \). This suggests that \( x = 6 \) is a point of local maximum for \( f \).

Final Answer

C. \( f \) has a relative maximum at \( x = 6 \) but not an absolute maximum.

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